Elastic & Inelastic Collision Simulator

Two carts, blue m₁ on the left and coral m₂ on the right, sit on a frictionless track. Positive velocity means moving to the right. The cart boxes are drawn larger for larger masses.

The table shows v₁, v₂, the total momentum p (kg·m/s) and the total kinetic energy KE (J), in a before column and an after column. Watch the p row first: it matches in both columns for every setting and either collision type. Then watch the KE row: it matches only for Elastic.

The after column is calculated as though the carts do collide. A collision only happens if cart 1 is moving to the right faster than cart 2 ($v_1 > v_2$); for other settings the carts never meet, so ignore the after values and the animation. The default settings (2 kg at 4 m/s and 3 kg at −1 m/s) give a clean case to start with: elastic gives −2 and +3 m/s with 17.5 J kept, while sticking gives 1 m/s and only 2.5 J.

Momentum is $\vec{p} = m\vec{v}$. With no net external force on the two-cart system, total momentum is conserved in every collision: $$m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$$

In a perfectly inelastic collision the carts share one final velocity: $$v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}$$ Kinetic energy, $K = \tfrac{1}{2}mv^2$, always decreases; the missing energy goes into deformation, sound and thermal energy.

In an elastic collision total kinetic energy is also conserved. Solving the two conservation equations together shows that the relative speed is unchanged and reversed, $v_{1i} - v_{2i} = -(v_{1f} - v_{2f})$, and gives $$v_{1f} = \frac{(m_1 - m_2)v_{1i} + 2m_2 v_{2i}}{m_1 + m_2} \qquad v_{2f} = \frac{(m_2 - m_1)v_{2i} + 2m_1 v_{1i}}{m_1 + m_2}$$ These two results are not on the AP equation sheet; on the exam you can start from conservation of momentum and kinetic energy.

Most real collisions fall between these two extremes: momentum is conserved but only some kinetic energy is lost.

A 1 kg cart moving at 5 m/s hits a 4 kg cart at rest. Find the final velocities for an elastic collision and for a collision in which the carts stick.

Total momentum. $p = (1)(5) + (4)(0) = 5$ kg·m/s, before and after in both cases. Initial $K = \tfrac{1}{2}(1)(5)^2 = 12.5$ J.

Elastic. $v_{1f} = \dfrac{(1 - 4)(5)}{5} = -3$ m/s and $v_{2f} = \dfrac{2(1)(5)}{5} = 2$ m/s. The light cart bounces back. Check momentum: $(1)(-3) + (4)(2) = 5$ kg·m/s. Check energy: $\tfrac{1}{2}(1)(9) + \tfrac{1}{2}(4)(4) = 4.5 + 8 = 12.5$ J.

Stick together. $v_f = 5/5 = 1$ m/s, so $K_f = \tfrac{1}{2}(5)(1)^2 = 2.5$ J. The collision turns 10 J, or 80% of the kinetic energy, into other forms.

Impulse. In the elastic case the 4 kg cart gains 8 kg·m/s of momentum and the 1 kg cart's momentum changes from +5 to −3 kg·m/s, a change of −8 kg·m/s. Equal and opposite impulses are Newton's third law at work.

In the simulator set m₁ = 1, m₂ = 4, v₁ = 5 and v₂ = 0. In Elastic mode the after column reads −3 and 2 with KE 12.5 J in both columns; switch to Inelastic (stick) and it reads 1 and 1 with KE 2.5 J.

Collisions are the core of Unit 4 (Topic 4.4, Elastic and Inelastic Collisions). Expect to classify a collision from before-and-after data by checking whether total kinetic energy changes, to find a final velocity from momentum conservation, and to explain a momentum graph or bar chart. Collisions also show up joined to other topics, such as a block that sticks to a pendulum and then swings up, where momentum handles the collision and energy handles the swing.